Vaccination is usually modelled by placing an individual into a separate vaccinated compartment, such that vaccinated nodes block incoming infections, so that if enough people are vaccinated the infection can no longer spread through the population, a phenomena referred to as herd immunity. This assumes that a vaccine provides full protection against transmission of an infectious disease for any individual. In general, it may be desirable to relax this assumption. For example, although the efficacy of a vaccine in preventing symptoms is ascertained before approval for public use, it is more difficult to determine how a given vaccine prevents transmission until data is collected during or after vaccine roll-out. In this case, it may be desirable to assume that vaccination reduces the transmissibility of an individual to a small but finite value. Furthermore, it is also important to recognise that vaccines are usually given to priority groups first, such as people with underlying health conditions, medical staff, or people above a certain age. Due to the correlated nature of social contacts [1]}, [2]}, prioritising vaccinations may lead to correlations between the vaccinated status of a node and its topological properties. Variation in the transmissibility of individuals has been explored in previous works [3]}, [4]}, [5]}, [6]}, however, analytical results have been restricted to the average risk or size of the epidemic. It has previously been shown that there is high variability in the risk of individual nodes, due to differences in the node environment [7]}, [8]}, which is neglected when only considering the average risk. In our work, we show that this variability is more prominent when transmissibility varies between individuals and focus on how this impacts the distribution of risk.
